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Current Question (ID: 17289)

Question:

$\text{A resistance of } 20 \, \Omega \text{ is connected to a source of an alternating potential, } V = 220 \sin(100 \pi t).$ $\text{The time taken by the current to change from its peak value to its rms value will be:}$

Options:

1. $0.2 \, \text{sec}$
2. $0.25 \, \text{sec}$
3. $25 \times 10^{-3} \, \text{sec}$
4. $2.5 \times 10^{-3} \, \text{sec}$

Solution:

$(4) \text{ Peak value to } r.m.s. \text{ value means current becomes } \frac{1}{\sqrt{2}} \text{ times.}$ $\text{Time taken by the current to reach its peak value:}$ $i = i_0 \sin 100 \pi t \Rightarrow i_0 = i_0 \sin 100 \pi t_1 \Rightarrow \sin 100 \pi t_1 = \sin \frac{\pi}{2}$ $\Rightarrow t_1 = \frac{1}{200} \, \text{sec}$ $\text{Time taken by the current to become } \frac{i}{\sqrt{2}} \text{ after reaching its peak value:}$ $i = i_0 \sin 100 \pi t \Rightarrow \frac{1}{\sqrt{2}} \times i_0 = i_0 \sin 100 \pi t_2 \Rightarrow \sin 100 \pi t_2 = \sin \frac{3\pi}{4}$ $\Rightarrow t_2 = \frac{3}{400} \, \text{sec}$ $\text{Time taken by the current to change from its peak value to r.m.s. value:}$ $t_2 - t_1 = \frac{3}{400} - \frac{1}{200} = \frac{1}{400} \, \text{sec} = 2.5 \times 10^{-3} \, \text{sec}$

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Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}